The invertible matrix theorem

The invertible matrix theorem

Lessons

The Invertible Matrix Theorem states the following:
Let AAA be a square n×nn \times nn×n matrix. Then the following statements are equivalent. That is, for a given AAA, the statements are either all true or all false.

1. AAA is an invertible matrix.

2. AAA is row equivalent to the n×nn \times nn×n identity matrix.

3. AAA has nnn pivot positions.

4. The equation Ax=0Ax=0Ax=0 has only the trivial solution.

5. The columns of AAA form a linearly independent set.

6. The equation Ax=bAx=bAx=b has at least one solution for each bbb in Rn\Bbb{R}^nR​n​​.